Research Article Asymptotic Distribution of Isolated Nodes in Secure Wireless Sensor Networks under Transmission Constraints

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1 Discrete Dynamics in Nature and Society Volume 16, Article ID , 1 pages Research Article Asymptotic Distribution of Isolated Nodes in Secure Wireless Sensor Networks under Transmission Constraints Y. Tang 1 and Q. L. Li 1 College of Culture and Communication, Changsha Social Work College, Changsha 4119, China College of Mathematics and Computer Science, Hunan Normal University, Changsha 4181, China Correspondence should be addressed to Y. Tang; cstangyan@16.com Received 18 December 15; Revised 5 May 16; Accepted 6 June 16 Academic Editor: Gabriella Bretti Copyright 16 Y. Tang and Q. L. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Eschenauer-Gligor (EG) key predistribution is regarded as a typical approach to secure communication in wireless sensor networks (WSNs). In this paper, we establish asymptotic results about the distribution of isolated nodes and the vanishing small impact of the boundary effect on the number of isolated nodes in WSNs with the EG scheme under transmission constraints. In such networks, nodes are distributed either Poissonly or uniformly over a unit square. The results reported here strengthen recent work by Yi et al. 1. Introduction A wireless sensoetwork is composed of a collection of wireless sensors distributed over a geographic region. A wireless sensoetwork can be an integral part of military command, control, communication, computing, intelligence, surveillance, reconnaissance, and target system. It has been the subject of intense research in recent decades. Asymptotic analysis, valid when the number of nodes in the network is large enough, has been useful for understanding the characteristic of the network. In many applications, a large number of wireless sensors are independently and uniformly deployed in the sensor field. They can be deployed by dropping from a plane or delivered in a missile. To model such a randomly deployed wireless sensoetwork, it is natural to represent the sensoodes by a finite random point process over a network square area S. In addition, due to the short transmission range of communication links, two wireless sensors can build a link if and only if they are within each other s transmission range. Assume all n sensors have the same transmission range of radius, then the induced network topology is a -disk graph in which two nodes are joined by an edge if and only if their distance is at most.thismodelisproposedbygilbert [1] and referred to as a random geometric graph, denoted by G(n,, S). However, in many applications, a wireless sensoetwork is composed of low cost sensors. Due to the ited capacity, traditional security schemes and key management algorithms aretoocomplicatedandnotfeasibleforsuchasystem.the Eschenauer-Gligor (EG) [] key predistribution scheme is a widely recognized way to secure communication. In this scheme,inawsnwithn sensors and sensor set V = {V 1, V,...,V n }, the EG scheme independently assigns a set of K n distinct cryptographic keys, which are selected uniformly at random from a pool of P n keys, to each sensoode; the set of keys of each sensor is called the key ring and is denoted by S x for sensor V x. The EG scheme is denoted by a random key graph G(n, K n,p n ) in [3 6], in which an edge exists between two nodes V x and V y if and only if they possess at least one common key; that is, S x S y =. In a WSN using the EG scheme under transmission constraint, two sensors V x and V y establish a direct link betweenthemifandonlyiftheyshareatleastonekey and their distance is no greater than. We denote the event establishing this direct link by E xy.ifweletthegraph G(n, θ n, S) model such a WSN, it is obvious to see G(n, θ n, S) is the intersection of random key graph G(n, K n,p n ) and

2 Discrete Dynamics in Nature and Society random geometric graph G(n,, S) with n nodes uniformly distributed over a square S;namely, G (n, θ n, S) =G(n, K n,p n ) G(n,, S), (1) where θ n represents the parameters K n,p n,and together. Let V x S be the location of point V x.adirectlinke xy exists in G(n, θ n, S) if both of the following two conditions are satisfied: V x V y ; S x S y =, where represents the Euclidean norm. We let p s be the probability of key sharing between two sensors and note that p s isalsotheedgeprobabilityinrandom key graph G(n, K n,p n ).Itholdsthat p s = P (S x S y () =). (3) Clearly if P n <K n,thenp s =1.IfP n K n,asshownin previous work [5 7], we have p s =1 ( P n K n K n ) ( P n K n ). (4) If P n K n,by[8],itfurtherholdsthat p s K n P n. (5) By [9], (5) implies that if K n /P n = o(1),then p s = K n (1 O( K n )) K n. (6) P n P n P n Wewilluse(5)and(6)throughoutthepaper. Let p e be the probability that a secure link exists between two sensors in the WSN with EG scheme under practical constraint; obviously, p e is the edge probability in G(n, θ n, S). It holds that p e = P[E xy ].Itissimplemattertoshowp e πr n p s and p e (1 ) πr n p s,if = o(1), p e πr n p s. Therefore if = o(1) and K n /P n = o(1), weget p e πr n K n /P n in view of (6). The secure wireless sensoetwork with nodes uniformly distributed is modeled by graph G(n, θ n, S), inwhichtwo nodes have an edge if their Euclidean distance is at most andhaveasecurelinkiftheirkeyringshaveatleastone common key. Compared to the work done by Yi et al. [1], in this paper, by using a different method, we establish that the number of isolated nodes in the secure wireless sensor network has an asymptotic Poisson distribution whether the n nodes are induced by a uniform point process or Poisson point process. To model transmission constraints, we use the popular disk model [11], and under the disk model, two nodes are directly connected if and only if their Euclidean distance is smaller or equal to a given threshold,whereparameter is termed as the transmission range. The following notations are used throughout the paper: (i) a(n) = o(b(n)) if n (a(n)/b(n)) ; (ii) a(n) = ω(b(n)) if n (b(n)/a(n)) ; (iii) a(n) b(n) if n (a(n)/b(n)) 1; (iv) a(n) = Θ(b(n)) if there exist sufficiently n and c 1,c > such that, for any n>n, c 1 b(n) a(n) c b(n); (v) a(n) = Ω(b(n)) if there exist sufficiently n and C> such that, for any n>n, a(n) Cb(n); (vi) the notation ln stands for the natural logarithm function. The rest of the paper is organized as follows. Section reviews related work. Section 3 comparatively studies the distribution of isolated nodes in WSNs employing the EG scheme with nodes Poissonly or uniformly distributed over a unit square S. Through the study, it shows that under certain conditions the impact of boundary effect on the number of isolated nodes is negligible. Finally, Section 4 summarizes conclusions and discusses prospects of establishing the distribution of isolated nodes and tighter connectivity thresholds for secure wireless sensoetworks under improved conditions.. Related Work With regard to the sensor distribution, we consider that n nodes are independently and uniformly deployed in unit square area S. The disk model induces a random geometric graph [4, 1 15] which is denoted by G(n,, S); anedge exists between two sensors if and only if their Euclidean distance is no more than. Extensive research has been done on random geometric graphs. The connectivity of random geometric graphs has been studied by Dette and Henze [16], Penrose[17],andothers[1,18 ].Forauniformn point process over a unit area square S, Dette and Henze [16] showed that, for any constant c, if = (ln n+c)/πn,then graph G(n,, S) has no isolated nodes with probability e c asymptotically. Eight years later, Penrose [14] established that if a random geometric graph induced by a uniform point process or Poisson point process has no isolated nodes, then it is almost surely connected. Besides the overall connectivity, some applications are concerned with whether there exists a giant connected component. Continuum percolation is a useful theorem in analyzing threshold phenomena. AmmariandDas[1]focusedonpercolationincoverage and connectivity in three-dimensional space and found out whether the network provides long distance multihop communication. For random key graph G(n, K n,p n ), Godehardt and Jaworshi[]focusedonthedistributionofthenumber of isolated vertices in G(n, K n,p n ). Some partial results concerning the connectivity of random key graphs were given in [6, 7, 3]. In [5], Rybarczyk gave asymptotic tight bounds for the thresholds of the connectivity, phase transition, and diameter of the largest connected component in random key graphs for all ranges of K n.otherrelated works regarding G(n, K n,p n ) model have been reported.

3 Discrete Dynamics in Nature and Society 3 For example, Bloznelis et al. [4] treated the evolution of the order of the largest component. Connectivity and communication security aspects of G(n, K n,p n ) in various important settings are also studied in [7, 5, 6]. Although some properties of secure WSNs with the EG scheme have been extensively studied in [4 7, 13, 7], most research [5 7, 7] unrealistically assumes unconstrained sensor-to-sensor communications; that is to say, any two sensors can communicate regardless of the distance between them. Recently, there is interest in random graphs in which an edge is determined by more than one random property, that is, intersection of different random graphs. The intersection of Erdős-Rényi random graph G(n, p) [8] and random geometric graph G(n,, S) has been of interest for quite some time now. Recent work on such random graphs is by [, 9] where connectivity properties and the distribution of isolated nodes are analyzed. And the intersection of random graphs G(n, p) and random key graph G(n, K n,p n ) is considered in [3]. Such a graph is constructed as follows: arandomkeygraphg(n, K n,p n ) is first formed based on the key distribution and each edge in this graph is deleted with a specified probability. The intersection of random key graph G(n, K n,p n ) and random geometric graph G(n,, S) (i.e., G(n, θ n, S)) isfirst studied in [31]. Di Pietro et al. [31] have shown that, under the scaling πr n (K n /P n) = c(ln n/n), the one law that this classofrandomgraphsisconnectedfollowsif > and c > π. Anotheotable work is due to Krzywdziński and Rybarczyk [13], where the authors have improved this result and established that, in G(n, θ n, S), ifπr n (K n /P n) = 8(ln n/n) with K n >, P n = ω(1), andwithoutany constraint on,theng(n, θ n, S) is almost surely connected. AndKrishnanetal.[4]demonstratedthatifπr n (K n /P n) π(ln n/n) with K n = ω(1) and (K n /P n) = o(1), then G(n, θ n, S) is almost surely connected. Recently, Tang and Li[3]andZhaoetal.[33]presentedthefirstzero-one laws for connectivity in G(n, θ n, S); these laws improve theresults[4,13]significantlyandhelpspecifythecritical transmission ranges for connectivity. Also the distribution of isolated nodes in G(n, θ n, S) is considered by Yi et al. [1] andpishro-niketal.[34],wherethenetworkwithn nodes distributed uniformly over a unit disk D or a unit square S. In addition to random key graphs and random geometric graphs, the Erdös-Rényi graph G(n, p) [8] has also been extensively studied. An Erdös-Rényigraph G(n, p) is defined on a set of n nodes such that any two nodes establish an edge independently with probability p. Asalreadyshown in the literature [5 7, 7], random key graph G(n, K n,p n ) and Erdös-Rényi graph G(n, p) have similar connectivity properties when they are matched through edge probability; that is, p s = p. Hence, it would be tempting to exploit this analogue and conclude that the distribution of isolated nodes in G(n, θ n, S) (G(n,K n,p n ) G(n,, S)) is similar to that of G(n, p) G(n,, S) in [], whether the n nodes are distributed Poissonly or uniformly on a unit square S. 3. Main Result In this section, we study the expected number of isolated nodes in WSNs with the EG scheme under transmission constraints with nodes either Poissonly or uniformly on a unit square S. The number of isolated nodes is a key parameter in the analysis of network connectivity. A necessary condition for a network to be connected is that the network has no isolated nodes, and this may be possibly true for the intersection of random key graph and random geometric graph. In order to obtain the distribution of isolated nodes in G(n, θ n, S), we prove the same result for its Poissonized version, graph G Poisson (n, θ n, S), where the only difference between G Poisson (n, θ n, S) and G(n, θ n, S) is that the node distribution of the former is a homogeneous Poisson point process with intensity n on a unit square S whilethatofthe latter is a uniform n point process Expected Number of Isolated Nodes in G Poisson (n, θ n, S). In graph G Poisson (n, θ n, S),letI x denote the event that node V x is isolated, and let D rn ( V x ) denote the intersection of S and the disk centered at position V x S with radius.whennode V x is at position V x, the number of nodes within area D rn ( V x ) follows a Poisson distribution with mean nd rn ( V x ),andto have an edge with V x in graph G Poisson (n, θ n, S), a node not only has to be within D rn ( V x ) butalsohastoshareatleastakey with node V x. Then the number of nodes neighboring to V x at V x follows a Poisson distribution with mean np s D rn ( V x ),and the probability that such number is is equal to e np s D ( V x). Integrating V x over S, then the probability that the node V x is isolated is given by P (I x )= S. (7) Theorem 1. Suppose that K n /P n = o(1), p s = ω(1/ ln n), and n nodes are Poissonly distributed on a unit square S with the maximum transmission radius πr n K n /P n =(ln n+c)/nfor some constant c. Then the expected number of isolated nodes in G Poisson (n, θ n, S) converges asymptotically to e c as n. Proof. Let W denote the number of isolated nodes in graph G Poisson (n, θ n, S).By(7),weknowP(I x )= S holds. To compute P(I x ) based on S, wedivides in a way similar to that by Li et al. [35] and Wan and Yi [15]. Specially, S is divided into S, S 1, S,andS 3,respectively,as illustrated in Figure 1 (note that = o(1)). S consists of all points each with a distance greater than to its nearest edge to S,whereasS 3 is a square of size at the four corners of S. We further divide S\{S S 3 } into S 1 and S as follows. In S\{S S 3 }, S 1 contains points whose distance to the nearest edge of S is no greater than /, while the remaining area is S. Then the expected number of isolated nodes is given by E (W) = np (I n n x)= n e n S np s D ( V x) d V x = n n S

4 4 Discrete Dynamics in Nature and Society 1 Since S 1 consists of four retangles, each of which has length 1 and width /,itfollowsthat 1 S 3 S S 1 S S S 1 S S 1 S 3 n n S 1 =4n(1 ) / n For simplicity, we write H(g) as H.Then e np sh(g) dg. (13) S e np sh(g) dg = (np s ) 1 (H ) 1 de np sh = (np s ) 1 S 3 S r 3 n S 1 {d[(h ) 1 e np sh ] e np sh d(h ) 1 }=(np s ) 1 (14) {d[ (H ) 1 e np sh ] (H ) H e np sh dg}. Figure 1: The unit square S and its divisions S, S 1, S,andS 3. + n n S 1 In view of (14), we further have (H ) H e np sh dg = (H ) 3 H e np sh dh = (np s ) 1 (H ) 3 H d( e np sh ). (15) + n n S + n e np s D ( V x) rn d V n x. S 3 (8) The four summands in (8) represent, respectively, the expected number of isolated nodes in the central area S,in the boundary area along the four sides of S, andinthefour corners of S. In the following analysis, we will show that the first term approaches e c as n,andtheremainingterms approach an n. Consider the first summand in (8). It is clear that, for any position V x S,wehave D rn ( V x ) = πr n, and note that πr n (K n /P n)=(ln n+c)/n, S = (1 ) to get n e np s D ( V x) d V n x = n e np sπ d V S n x S = n ne np sπr n S d V x = n e c (1 ) =e c. For the second term in (8), we introduce some notation as follows. For any position V x S 1, we let the distance from V x to the nearest edge of square S be g,where g /.For V x S 1,clearly D rn ( V x ) is determined by g, and we denote it by H(g).Itiseasytoobtain H(g)= D ( V x ) =πr n r n arccos g +g r n g, (9) (1) H (g) = r n g, (11) H g (g) =. rn g (1) For g /, it holds from (11) and (1) that H (H ) 3 = g 4(rn g ) / 4 ((3/4) r = n ) 9rn 3. (16) By (15) and (16), it follows that / (H ) H e npsh dg (17) rn/ 9rn 3p sn d( e npsh ). Applying (17) into (14), / e npsh(g) dg = 1 / ( d[ (H ) 1 e npsh ] np s / (H ) H e npsh dg) 1 / ( d[ (H ) 1 e npsh ] np s rn/ + 9rn 3p sn + e np sh() 9r 3 n p s n. d[ e np sh ]) e np sh() np s H () (18) From (1) and (11), we obtain H() = πr n / and H () =. Consider / e npsh(g) dg e np sh() np s H () + e npsh() 9rn 3p s n = e np s(πrn /) np s + e nps(π /) 9rn 3. p s n (19)

5 Discrete Dynamics in Nature and Society 5 Using πr n (K n /P n)=(ln n+c)/nand (6) in (19), we derive / Using () in (13), we get e npsh(g) dg e np s(πrn /) (1+o(1)). () np s n n S 1 =4n(1 ) 4n / n e nps(πr n /) n e np sh(g) dg np s (1+o(1)). (1) From K n /P n = o(1), p s = ω(1/ ln n), andwithδ denoting πr n p sn (note that Δ=Θ(ln n)), We obtain =π 1/ p 1/ s n 1/ Δ 1/. () n n S 1 =. (3) Now we consider n n S.For V x S, when the distance from V x to the nearest edge of square S is g,where / g, the function expression of D rn ( V x ) still is H(g). H(g) is increasing with g for / g by H (g) = rn g. Thus, when the distance from V x to the nearest edge of S equals /, thearea D rn ( V x ) reaches its minimum value: D ( V x ) min =H( )=( π )πr n. (4) With C denoting /3 + 3/4π, then D rn ( V x ) min =C πr n. And with S = (1 ), we derive n e np s D ( V x) d V n x n e np sc π d V S n x S = n ne np sc πr n S d V x n ne np sc πr n. From p s = ω(1/ ln n), Δ=Θ(ln n),and(),itfollows (5) n n S =. (6) For the last term in (8), we have (1/4)πr n D ( V x ) πr n for any V x S 3 and S 3 =4r n to get n e np s D ( V x) d V n x n e (1/4)np sπ d V S n x 3 S 3 = n ne (1/4)np sπr n S 3 d V x = n 4nr n n 1/4 e (1/4)np sπr n =, (7) where (), Δ = Θ(ln n), andp s = ω(1/ ln n) are used in reaching (7). As a result of (8),(9), (3), (6), and (7), we prove n =e c. (8) The parameter c in Theorem 1 is a constant, or it can depend on n, inwhichcasec = o(ln n). Thefollowing corollary can be established. Corollary. In graph G Poisson (n, θ n, S) under conditions K n /P n = o(1), p s = ω(1/ ln n),andπr n (K n /P n)=(ln n+c)/n with c=o(ln n) or a constant c, then for any constant ε> such that E(W) = o(n ε ) holds. Now we examine the impact of boundary effect on the number of isolated nodes in G Poisson (n, θ n, S) since the network area in our analysis is a square. The square accounts for the real-world boundary effect whereby some transmission region of a sensor close to the network boundary may fall outside the network field. In contrast, the torus einates the boundary effect. The analysis of impact of the boundary effectisdonebycomparingthenumberofisolatednodesin G Poisson (n, θ n, S) and the number in a network with nodes Poissonly distributed on a unit torus S T with a pair of nodes V x, V y separated by a toroidal distance V x V y T.Denotesuch network on a unit torus by G T Poisson (n, θ n, S T ).Thefollowing Theorem can be established. Theorem 3. Suppose that K n /P n = o(1), p s = ω(1/ ln n),and n nodes are Poissonly distributed on a unit torus S T with the maximum transmission radius πr n (K n /P n)=(ln n+c)/nfor some constant c. Then the expected number of isolated nodes in G T Poisson (n, θ n, S T ) converges to e c as n. Proof. Let W T denote the number of isolated nodes in graph G T Poisson (n, θ n, S T ), and let D T ( V x ) denote the intersection of S T and the disk centered at position V x S T with radius.theprobabilitythatnodev x is isolated is P(I T x )= S T e np s D T rn ( V x) d V x.sinces T isaunittorus,itholdsthat D T ( V x ) = πr n for any V x S T.Then P (I T x )= S T e np s D T rn ( V x) d V x =e πr n p sn S T d V x =e πr n p sn ST =e πr n p sn. Using the condition πr n (K n /P n)=(ln n+c)/n,weobtain (9) n E(WT )= np (I T n x )= n ne πr n psn =e c. (3) OnthebasisofTheorems1and3,usingthecoupling technique, the following theorem can be obtained. Theorem 4. Suppose that K n /P n = o(1), p s = ω(1/ ln n), and n nodes are Poissonly distributed on a unit square S

6 6 Discrete Dynamics in Nature and Society with the maximum transmission radius πr n (K n /P n) = (ln n+c)/nfor some constant c. Thenthenumberofisolated nodes in G Poisson (n, θ n, S) due to the boundary effect converges asymptotically to as n. Proof. Comparing Theorems 1 and 3, it is noted that the expected numbers of isolated nodes on a unit torus S T and on a unit square S, respectively, asymptotically converge tothesameconstante c as n.nowweusethe coupling technique [36] to construct the connection between W and W T. Consider a graph G T Poisson (n, θ n, S T ),andthe number of isolated nodes in that graph is W T.Remove each connection of the above graph with probability 1 D rn ( V x ) / D T ( V x ), independently of the event that another connection is removed. Due to D rn ( V x ) D T ( V x ), then 1 D rn ( V x ) / D T ( V x ) 1. We furtheote that only connections between nodes near the boundary will be affected. Denote the number of newly appearing isolated nodes by W E ;namely,w E is the number of isolated nodes due to the boundary effect; it is straightforward to show that W E is a nonnegative random integer. Further, such a connection removal process results in a random network with nodes Poissonly distributed with density n.thatis,arandom network on a unit square with boundary effect is included. The following equation result holds: W=W E +W T. (31) By Theorems 1 and 3 and the above equation, it can be shown that n E (WE ) = n E (W W T ) =. (3) Due to the nonnegativity of W E, n P (WE =)=1. (33) 3.. Distribution of the Number of Isolated Nodes in G Poisson (n, θ n, S). In this subsection, we analyze the distribution of the number of isolated nodes in G Poisson (n, θ n, S). For this purpose, we give some definitions. Let Poi(λ) be a Poisson random variable with parameter λ. Let Γ beafinitesetofindicesandlet(i a ) a Γ be a family of random indicator variables. We say (I a ) a Γ are positively related(see[37]), if, for each a Γ, there exist random indicator variables (J ba ) b Γ\{a} with the distributions L ((J ba ) b Γ\{a} )=L ((I b ) b Γ\{a} I a =1), (34) such that J ba I b for every b =a. A useful result obtained by Stein-Chen method is the following. Lemma 5 (see [37, 38]). Suppose that Y = a Γ I a,where the indicator variables (I a ) a Γ are positively related random indicator variables. Then one has d TV (Y, Poi (EY)) 1 e EY EY (Var Y EY+ (EI a ) ). a Γ (35) For x = 1,...,n,letI T x = 1 if node V x is isolated in G T Poisson (n, θ n, S T ) and W T = n x=1 IT x. Therefore, WT is the number of isolated nodes in G T Poisson (n, θ n, S T ) as defined in Theorem 3. We will demonstrate the asymptotic Poisson distribution of W T by employing the Stein-Chen method [37]. The result is given in Theorem 6. Theorem 6. Suppose K n ln n/ ln ln n, K n /P n = o(1), p s = ω(1/ ln n),andn nodes are Poissonly distributed on a unit torus S T with the maximum transmission radius πr n (K n /P n)= (ln n+c)/nfor some constant c. Then the distribution of the number of isolated nodes in G T Poisson (n, θ n, S T ) converges to a Poisson distribution with mean e c as n. Proof. The triangular inequality for the total variation distance implies d TV (W T, Poi (e c )) d TV (W T, Poi (EW T )) +d TV (Poi (EW T ),Poi (e c )). By a coupling argument [39] and Theorem 3, we have (36) d TV (Poi (EW T ),Poi (e c )) EWT e c =o(1). (37) Combining this with (36), we now only need to prove n d TV (W T, Poi (EW T )) =. (38) First, we claim that (I T x )n x=1 are positively related. To see this, define I T yx =1 if node V y is isolated in G T Poisson (n, θ n, ST 1 ), (39) where G T Poisson (n, θ n, ST 1 ) is a graph with P n \ S x or S T 1 = ST \ D T ( V x ) compared to G T Poisson (n, θ n, S T ),wheres x represents the key ring which is adjacent to V x in G T Poisson (n, θ n, S T ). Conditional on the isolation of node V x in G T Poisson (n, θ n, S T ), any node V y is not adjacent to S x or V y S T \D T ( V x ) in G T Poisson (n, θ n, S T ).Hence,wehave L ((I T yx )n )=L y=1,y=x ((IT y )n y=1,y=x IT x =1). (4) For every y =x,ifi T y =1then IT yx =1.Consequently,weget I T yx IT y.

7 Discrete Dynamics in Nature and Society 7 By Lemma 5, the binary nature, and exchangeability of the random variables involved, we find that d TV (W T, Poi (EW T )) T 1 e EW EW T (Var W T EW T + (EI T x ) ) n x=1 1 EW T (Var WT EW T + (EI T x ) ) n x=1 1 EW T (n (n 1) E(IT x IT y ) n(n ) (EIT x ) ). (41) The cross term E(I T x IT y ) in (41) is given by [33], and we have E(I T x IT y ) (1 4πr n )e πr n p sn (4) +4πr n e πr n psn e πr n n(k4 /P n )+(K n /(P n K n ))e πr n n(k n/pn). With F(n) denoting e πr n n(k4 n /P n )+(K n /(P n K n ))e πr n n(k n/pn),combining (9), (41), and (4) readily gives d TV (W T, Poi (EW T )) n[(n 1) e πr n psn (1 + 4πr n F (n)) (n ) e πr n psn ] ne πr n p sn ne π psn (1 + 4nπr nf (n)) ne πr n p sn e πr n psn (1 + 4nπr nf (n)) =o(1), (43) where K n ln n/ ln ln n, K n /P n = o(1), p s = ω(1/ ln n) πr n (K n /P n)=(ln n+c)/n,and()areusedinreaching(43), along with (36), (37), and (43), which concludes the proof. We now consider the asymptotic distribution of the number of isolated nodes in G Poisson (n, θ n, S). From Theorem 4, n P(W E =)=1holds, and using Slutsky s Theorem [4], the following result on the asymptotic distribution of W canbereadilyobtained. Theorem 7. Suppose K n ln n/ ln ln n, K n /P n = o(1), p s = ω(1/ ln n), andn nodes are Poissonly distributed on a unit square S with the maximum transmission radius πr n (K n /P n) = (ln n + c)/n for some constant c. Then the distribution of the number of isolated nodes in G Poisson (n, θ n, S) converges to a Poisson distribution with mean e c as n Distribution of the Number of Isolated Nodes in G(n, θ n, S). We derive the distribution of isolated nodes in G(n, θ n, S) by using standard Poissonization technique [11, 14]. The idea is that the result about the distribution of isolated nodes for G(n, θ n, S) follows once we establish the resultwithpoissonization,thatistosay,onceweobtainthe distribution of isolated nodes for G Poisson (n, θ n, S). Seethe following lemma for rigorous argument. Lemma 8. Suppose that K n /P n = o(1), p s = ω(1/ ln n), and n nodes have the same maximum transmission radius πr n (K n /P n)=(ln n+c)/nfor some constant c. Thenwith m denoting n n 1/+c,wherec is an arbitrary constant with <c <1/, then node V i is isolated in G(n, θ n, S) if and only if V i is an isolated node in G Poisson (m, θ n, S). Proof. We will use the standard de-poissonization technique [11, 14, 18] to prove Lemma 8. Let M denote the number of nodes in graph G Poisson (m, θ n, S); clearlym follows a Poisson distribution with mean m, for any positive t, and from Chebyshev s inequality, we have P ( M m t m) t. (44) Without loss of generality, we take n n 1/+c as an integer. With t = (n m)/ m,substitutingm=n n 1/+c into (46), we get P (M n n 1/+c or M n) n n1/+c =o(1). n 1+c (45) Hence, n n 1/+c <M<nholds almost surely. When M<n,weconstructacouplingC between graph G(n, θ n, S) and graph G Poisson (m, θ n, S);letgraphG(n, θ n, S) be the result of adding n Mnodes uniformly distributed on S to graph G Poisson (m, θ n, S). LetV P be the node set of G Poisson (m, θ n, S); clearly, V P is a subset of V, wherevis the node set of G(n, θ n, S). In addition, it is straightforward to see that the edge set of G Poisson (m, θ n, S) is also a subset of that of G(n, θ n, S). Then under coupling C,graphG Poisson (m, θ n, S) is a subgraph of G(n, θ n, S). Let D X,D P denote the set of isolated nodes in G(n, θ n, S) and G Poisson (m, θ n, S), respectively. To prove Lemma 8, we show P (D X It is straightforward to see P (D X =D P ) P [(D P \D X =D P )=o(1). (46) =) (n n 1/+c + P [(D X \D P =) (n n 1/+c + P (M n n 1/+c or M n). (47)

8 8 Discrete Dynamics in Nature and Society By (47), we will prove P(D X P [(D P \D X =o(1), P [(D X \D P =o(1). =D P ) = o(1) if we can derive =) (n n 1/+c =) (n n 1/+c (48) (49) To prove (48) and (49), with n n 1/+c <M<n, we consider the coupling C under which G Poisson (m, θ n, S) is a subgraph of G(n, θ n, S). First, we consider (48) with n n 1/+c <M<n;event D P \D X =happens if and only if there exists at least one node V i such that V i D P and V i D X ;thatistosay,v i is isolated in G Poisson (m, θ n, S) but is not isolated in G(n, θ n, S). Then there exists at least one node V in V\V P such that V and V i are neighbors in G(n, θ n, S). Dueto V \ V P = n M < n 1/+c,notingthatp e is the edge probability in G(n, θ n, S), thenwithw P denoting the number of isolated nodes in G Poisson (m, θ n, S), as an easy consequence of the union bound, P [(D P \D X W P n 1/+c p e. =) (n n 1/+c (5) In order to apply Corollary to G Poisson (m, θ n, S), forn sufficiently large, the following condition holds: πr n K n = P n We demonstrate (51) in view of ln m+c m. (51) m πr n K n ln n+c (ln m+c) =m (ln m+c) P n n =(n n 1/+c ln n+c ) (ln (n n 1/+c )+c) n =(1 n c 1/ ) (ln n+c) ln n ln (1 n c 1/ ) c=o(1), (5) where < c < 1/. And sufficiently large n areusedin the final step. Then with (51), for any constant ε>,weuse Corollary to get W P =o(m ε )=o(n ε ). (53) Note that p e πr n p s,andπr n (K n /P n)=((ln n+c)/n). Then from (5), with ε satisfying <ε<1/ c,itfollows that P [(D P \D X o(n ε ) n 1/+c =) (n n 1/+c ln n+c n =o(1). (54) Second, in order to prove (49) with n n 1/+c <M<n, we consider event that D X \D P =occurs if and only if there exists at least one node V j such that V j D X and V j D P. With V j D X,thenV j is isolated in G(n, θ n, S),whichalong with V j D P leads to V j V P and V j V\V P.Letqdenote the probability that a node is isolated in G(n, θ n, S);itfollows via a union bound that P [(D X \D P n 1/+c q. =) (n n 1/+c (55) Since q = S, according to the proof of Theorem 1, it is easy to show q = Θ(ln n/n). Withq = Θ(ln n/n) and <c <1/,then P [(D X \D P =o(1). =) (n n 1/+c (56) In conclusion, since P(D X =) P(D P = ) P(D X = D P ), the above discussions lead to establish Lemma 8. Applying Theorem 7 and Lemma 8, we get the following theorem. Theorem 9. Suppose K n ln n/ ln ln n, K n /P n = o(1), p s = ω(1/ ln n), and n nodes are uniformly distributed on a unit square S with the maximum transmission radius πr n (K n /P n) = (ln n+c)/nfor some constant c. Thenthe distribution of the number of isolated nodes in G(n, θ n, S) converges to a Poisson distribution with mean e c as n. Noting that the number of isolated nodes in a network is a nonnegative integer, the following result can be obtained as a consequence of Theorems 7 and 9. Notice that, in formulating this result, we drop the assumption that c originally is a constantandallowitinsteadtoben-dependent. Corollary 1. Suppose K n /P n = o(1), p s = ω(1/ ln n), andn nodes are Poissonly (uniformly) distributed on a unit square S with the maximum transmission radius πr n (K n /P n) = (ln n+c n )/n for some c n. A necessary condition for graph G Poisson (n, θ n, S)(G(n, θ n, S)) to be asymptotically connected is c n. 4. Conclusion and Future Work Yi et al. [1] considered that a wireless ad hoc network consists of n nodes distributed independently and uniformly in a unit disk D or a unit square S. They used Brun s sieve to show that, for graph G(n, θ n, D) or G(n, θ n, S),ifπr n (K n /P n)=(ln n+ c)/n and K n /P n = ω(1/ ln n), the number of isolated nodes asymptotically follows a Poisson distribution with mean e c. Pishro-Nik et al. [34] also obtained such result on asymptotic Poisson distribution with condition K n /P n = ω(1/ ln n) generalized to K n /P n = Ω(1/ ln n). In this paper, we discuss the distribution of isolated nodes in WSNs employing the widely Eschenauer-Gligor key

9 Discrete Dynamics in Nature and Society 9 predistribution scheme under transmission constraint with nodes distributed either Poissonly or uniformly on a unit square S or a unit torus S T. Using the coupling technique, it is shown that the impact of the boundary effect on the number of isolated nodes vanishes small as n.we obtain that the numbers of isolated nodes in graph G(n, θ n, S) and graph G Poisson (n, θ n, S) asymptotically follow a Poisson distribution with mean e c with πr n (K n /P n)=(ln n+c)/n and K n /P n = ω(1/ ln n). In practice WSNs, K n is expected to be several orders of magnitude smaller than P n,soitoftenholdsthatk n /P n = o(1/ ln n). We believe that the following conjecture is true. Conjecture 11. Suppose that K n /P n = o(1), K n /P n = o(1/ ln n), and n nodes are uniformly (Poissonly) distributed on a unit square S with the maximum transmission radius πr n (K n /P n) = (ln n+c)/nfor some constant c. Then the distribution of the number of isolated nodes in graph G(n, θ n, S) (graph G Poisson (n, θ n, S)) converges to a Poisson distribution with mean e c as n. Connectivity in secure wireless sensoetworks under transmission constraints is another important subject. Although the significant improved conditions and results for asymptotic connectivity are presented by Krishnan et al. [4], Krzywdziński and Rybarczyk [13], Tang and Li [3], and Zhao et al. [33], we want to show that the connectivity in WSN with theegscheme(i.e.,g(n, θ n, S)) isexactlyanalogueofthe counterpartinclassicrandomgraphs. Conjecture 1. Suppose that K n /P n = o(1), K n /P n = o(1/ ln n),andπr n (K n /P n)=(ln n+c n )/n. (i) If c n,thenwithhighprobabilityg(n, θ n, S) is disconnected. (ii) If c n c, then the probability that G(n, θ n, S) is connected tends to e e c. (iii) If c n,thenwithhighprobabilityg(n, θ n, S) is connected. 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